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In mathematics, a circle bundle is a
fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
where the fiber is the
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
S^1. Oriented circle bundles are also known as principal ''U''(1)-bundles. In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relat ...
, circle bundles are the natural geometric setting for
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
. A circle bundle is a special case of a
sphere bundle In the mathematical field of topology, a sphere bundle is a fiber bundle in which the fibers are spheres S^n of some dimension ''n''. Similarly, in a disk bundle, the fibers are disks D^n. From a topological perspective, there is no difference betw ...
.


As 3-manifolds

Circle bundles over
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...
s are an important example of
3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...
s. A more general class of 3-manifolds is
Seifert fiber space A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle ( circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for ...
s, which may be viewed as a kind of "singular" circle bundle, or as a circle bundle over a two-dimensional
orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. D ...
.


Relationship to electrodynamics

The Maxwell equations correspond to an
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical co ...
represented by a
2-form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
''F'', with \pi^F being cohomologous to zero, i.e. exact. In particular, there always exists a
1-form In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to e ...
''A'', the
electromagnetic four-potential An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.Gravitation, J.A. W ...
, (equivalently, the
affine connection In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
) such that : \pi^F = dA. Given a circle bundle ''P'' over ''M'' and its projection :\pi:P\to M one has the
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same ...
:\pi^*:H^2(M,\mathbb) \to H^2(P,\mathbb) where \pi^ is the
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: i ...
. Each homomorphism corresponds to a Dirac monopole; the integer
cohomology group In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
s correspond to the quantization of the
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respectiv ...
. The
Aharonov–Bohm effect The Aharonov–Bohm effect, sometimes called the Ehrenberg–Siday–Aharonov–Bohm effect, is a quantum mechanical phenomenon in which an electrically charged particle is affected by an electromagnetic potential (φ, A), despite being confine ...
can be understood as the
holonomy In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geomet ...
of the connection on the associated line bundle describing the electron wave-function. In essence, the Aharonov–Bohm effect is not a quantum-mechanical effect (contrary to popular belief), as no quantization is involved or required in the construction of the fiber bundles or connections.


Examples

* The
Hopf fibration In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Ho ...
is an example of a non-trivial circle bundle. * The unit tangent bundle of a surface is another example of a circle bundle. * The unit tangent bundle of a non-orientable surface is a circle bundle that is not a principal U(1) bundle. Only orientable surfaces have principal unit tangent bundles. * Another method for constructing circle bundles is using a complex line bundle L \to X and taking the associated sphere (circle in this case) bundle. Since this bundle has an orientation induced from L we have that it is a principal U(1)-bundle. Moreover, the characteristic classes from Chern-Weil theory of the U(1)-bundle agree with the characteristic classes of L. * For example, consider the analytification X a complex plane curve \text\left( \frac \right). Since H^2(X) = \mathbb = H^2(\mathbb^2) and the characteristic classes pull back non-trivially, we have that the line bundle associated to the sheaf \mathcal_X(a) = \mathcal_(a)\otimes \mathcal_X has Chern class c_1 = a \in H^2(X).


Classification

The
isomorphism class In mathematics, an isomorphism class is a collection of mathematical objects isomorphic to each other. Isomorphism classes are often defined as the exact identity of the elements of the set is considered irrelevant, and the properties of the str ...
es of principal U(1)-bundles over a manifold ''M'' are in one-to-one correspondence with the
homotopy class In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...
es of maps M \to BU(1), where BU(1) is called the classifying space for U(1). Note that BU(1)= \mathbbP^\infty is the infinite-dimensional
complex projective space In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of ...
, and that it is an example of the
Eilenberg–Maclane space In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name ...
K(\mathbb,2). Such bundles are classified by an element of the second integral cohomology group H^2(M,\mathbb) of ''M'', since : ,BU(1)\equiv ,\mathbb CP^\infty\equiv H^2(M). This isomorphism is realized by the
Euler class In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of ...
; equivalently, it is the first
Chern class In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ...
of a smooth complex
line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organising ...
(essentially because a circle is homotopically equivalent to \mathbb^*, the complex plane with the origin removed; and so a complex line bundle with the zero section removed is homotopically equivalent to a circle bundle.) A circle bundle is a principal U(1) bundle if and only if the associated map M \to B\mathbb Z_2 is null-homotopic, which is true if and only if the bundle is fibrewise orientable. Thus, for the more general case, where the circle bundle over ''M'' might not be orientable, the isomorphism classes are in one-to-one correspondence with the
homotopy class In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...
es of maps M \to BO_2. This follows from the extension of groups, SO_2 \to O_2 \to \mathbb Z_2, where SO_2 \equiv U(1).


Deligne complexes

The above classification only applies to circle bundles in general; the corresponding classification for smooth circle bundles, or, say, the circle bundles with an
affine connection In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
requires a more complex cohomology theory. Results include that the smooth circle bundles are classified by the second Deligne cohomology H_D^2(M, \mathbb); circle bundles with an affine connection are classified by H_D^2(M, \mathbb(2)) while H_D^3(M, \mathbb) classifies line bundle
gerbe In mathematics, a gerbe (; ) is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. They can be seen as an analogu ...
s.


See also

*
Wang sequence In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they hav ...


References

* . {{Manifolds Circles Fiber bundles K-theory